BA46 TAXIING FOR DEP AT CLE CROSSED RWY WITHOUT ATC CLRNC.

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I HAD JUST FINISHED A CABIN PA ANNOUNCEMENT AND HAD FAILED TO CHANGE MY COM PANEL TO MONITOR RADIO #1. I DID NOT HEAR THE TAXI CLRNC THE FO COPIED. I WAS SURE THAT I HEARD HIM READ BACK 'TAXI TO RWY 6C.' WE EXECUTED THAT INSTRUCTION. WE CALLED TWR WHILE HOLDING SHORT OF RWY 6C. I BELIEVE IT WAS THE SAME CTLR WORKING GND; TWR AND CLRNC DELIVERY. HE ANSWERED OUR CALL BY ASKING; 'DO YOU KNOW WHAT YOU JUST DID?' WE WERE MOMENTARILY CONFUSED; AS WE WERE CERTAIN THAT WE HAD EXECUTED THE CLRNC RECEIVED AND READ BACK. IT QUICKLY DAWNED ON US THE CTLR WAS OF A DIFFERENT OPINION. HE CLRED US TO DEPART RWY 6C. THERE WERE NO APPARENT CONFLICTS WITH OTHER TFC. I HAD VISUALLY CHKED THE FINAL APCH TO RWY 6R BEFORE PROCEEDING TO CROSS. I AM NOT SURE THAT THE CTLR SENT THE WRONG OR AN UNCLR TAXI INSTRUCTIONS; OR IF THE FO HAD HEARD WHAT HE EXPECTED TO HEAR; GIVEN THE DEP ATIS. I BELIEVE THAT HE READ BACK WHAT WE DID (TAXI TO RWY 6C); WITH NO CORRECTION FROM THE CTLR. I KNOW FOR CERTAIN THAT I MADE AN ERROR BY NOT BACKING UP THE FO ON THE RADIO AS HE COPIED THE TAXI CLRNC. THIS WAS UNINTENTIONAL.